Nuprl Lemma : apply-alist-function
∀[T,A:Type]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[F:T ⟶ A]. ∀[L:T List].
apply-alist(eq;map(λx.<x, F[x]>;L);x) = (inl F[x]) ∈ (A?) supposing (x ∈ L)
Proof
Definitions occuring in Statement :
apply-alist: apply-alist(eq;L;x),
l_member: (x ∈ l),
map: map(f;as),
list: T List,
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
unit: Unit,
lambda: λx.A[x],
function: x:A ⟶ B[x],
pair: <a, b>,
inl: inl x,
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
not: ¬A,
false: False,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
apply-alist: apply-alist(eq;L;x),
top: Top,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
pi1: fst(t),
pi2: snd(t),
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
eqof: eqof(d),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
list_induction,
isect_wf,
l_member_wf,
equal_wf,
unit_wf2,
apply-alist_wf,
map_wf,
list_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
cons_member,
map_cons_lemma,
list_ind_cons_lemma,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
and_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
cons_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesisEquality,
hypothesis,
unionEquality,
productEquality,
independent_pairEquality,
applyEquality,
functionExtensionality,
inlEquality,
independent_functionElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
voidElimination,
lambdaFormation,
rename,
dependent_functionElimination,
productElimination,
isect_memberEquality,
voidEquality,
setElimination,
unionElimination,
equalityElimination,
dependent_set_memberEquality,
independent_pairFormation,
applyLambdaEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
axiomEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T,A:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:T]. \mforall{}[F:T {}\mrightarrow{} A]. \mforall{}[L:T List].
apply-alist(eq;map(\mlambda{}x.<x, F[x]>L);x) = (inl F[x]) supposing (x \mmember{} L)
Date html generated:
2017_04_14-AM-08_46_44
Last ObjectModification:
2017_02_27-PM-03_33_38
Theory : list_0
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